One option-pricing problem that has hitherto remained unsolved is the pricing of a European call on a stock that has a stochastic volatility. From the work of Merton [12], Garman [6], and Cox, Ingersoll, and Ross [3], the differential equation that the option must satisfy is known. The solution of this differential equation is independent of risk preferences if (a) the volatility is a traded asset or (b) the volatility is uncorrelated with aggregate consumption. If either of these conditions holds, the risk-neutral valuation arguments of Cox and Ross [4] can be used in a straightfoward way.

This paper produces a solution in series form for the situation in which the stock price is instantaneously uncorrelated with the volatility. We do not assume that the volatility is a traded asset. Also, a constant correlation between the instantaneous rate of change of the volatility and the rate of change of aggregate consumption can be accommodated. The option price is lower than the Black-Scholes (B-S) [1] price when the option is close to being at the money and higher when it is deep in or deep out of the money. The exercise prices for which overpricing by B-S takes place are within about ten percent of the security price. This is the range of exercise prices over which most option trading takes place, so we may, in general, expect the B-S price to overprice options. This effect is exaggerated as the time to maturity increases. One of the most surprising implications of this is that, if the B-S equation is used to determine the implied volatility of a near-the-money option, the longer the time to maturity the lower the implied volatility. Numerical solutions for the case in which the volatility is correlated with the stock price are also examined.

The stochastic volatility problem has been examined by Merton [13], Geske [7], Johnson [10], Johnson and Shanno [11], Eisenberg [5], Wiggins [16], and Scott [15]. The Merton and Geske papers provide the solution to special types of stochastic volatility problems. Geske examines the case in which the volatility of the firm value is constant so that the volatility of the stock price changes in a systematic way as the stock price rises and falls. Merton examines the case in which the price follows a mixed jump-diffusion process. Johnson [10] studies the general case in which the instantaneous variance of the stock price follows some stochastic process. However, in order to derive the differential equation that the option price must satisfy, he assumes the existence of an asset with a price that is instantaneously perfectly correlated with the stochastic variance. The existence of such an asset is sufficient to derive the differential equation, but Johnson was unable to solve it to determine the option price. Johnson and Shanno [11] obtain some numerical results using simulation and produce an argument aimed at explaining the biases observed by Rubinstein [14]. Eisenberg [5] examines how options should be priced relative to each other using pure arbitrage arguments. Numerical solutions are attempted by Wiggins [16] and Scott [15].

Section I of this paper provides a solution to the stochastic volatility option-pricing problem in series form. Section II discusses the numerical methods that can be used to examine pricing biases when the conditions necessary for the series solution are not satisfied. Section III investigates the biases that arise when the volatility is stochastic but when a constant volatility is assumed in determining option prices. Conclusions are in Section IV.